Chapter 1 – Real Numbers | CBSE Class 10 Basic Mathematics 2025

I

Multiple Choice Questions

1. HCF / LCM Property

1 Mark

If the HCF of two positive integers \(a\) and \(b\) is 1, then their LCM is:

(A) \(a+b\)

(B) \(a\)

(C) \(b\)

(D) \(ab\)

Set 430/1/1

2. Arithmetic Calculation

1 Mark

The value of \((\text{HCF}-\text{LCM})\) for the two numbers 3 and 5 is:

(A) 2

(B) 4

(C) 14

(D) \(-14\)

Set 430/2/1

3. Ending Digits

1 Mark

The number \(2^n\), where \(n\) is a natural number, cannot end with the digit:

(A) 4

(B) 6

(C) 2

(D) 0

Set 430/2/1

4. Finding Unknown Variable

1 Mark

If \(\operatorname{HCF}(x,20)=2\) and \(\operatorname{LCM}(x,20)=60\), then the value of \(x\) is:

(A) 3

(B) 6

(C) 20

(D) 10

Set 430/4/1

5. Exponents in HCF / LCM

1 Mark

If \(p=2^3\times3^2\times5\) and \(q=2^2\times3^3\), then the LCM of \(p\) and \(q\) is:

(A) \(2^3\times3^3\)

(B) \(2^2\times3^2\)

(C) \(2^2\times3^2\times5\)

(D) \(2^3\times3^3\times5\)

Set 430/3/1

6. Definition Based

1 Mark

A prime number has:

(A) exactly two prime factors

(B) exactly one prime factor

(C) at least one prime factor

(D) at least two prime factors

Set 430/3/1

7. Simplification Type

1 Mark

\[ (2-5\sqrt3)^2 \] is:

(A) a negative integer

(B) an irrational number

(C) a rational number

(D) a positive integer

Set 430/5/1

II

Assertion – Reason Questions

8. HCF and LCM Relation

1 Mark

Assertion (A): For any two natural numbers \(a\) and \(b\), the HCF of \(a\) and \(b\) is a factor of the LCM of \(a\) and \(b\).

Reason (R): HCF of any two natural numbers divides both the numbers.

Set 430/1/1

9. Prime Divisibility

1 Mark

Assertion (A): The prime numbers which divide 36 also divide 6.

Reason (R): Any number which divides \(p^2\) also divides \(p\).

Set 430/2/1

10. Product of Irrational Expressions

1 Mark

Assertion (A): \[ (a+\sqrt b)(a-\sqrt b) \] is a rational number, where \(a\) and \(b\) are positive integers.

Reason (R): Product of two irrationals is always rational.

Set 430/6/1

III

Very Short Answer Type Questions

11. Proof of Non-ending Zero

2 Marks

Show that \(45^n\) cannot end with the digit 0, where \(n\) is a natural number.
Write the prime number \(a\) which on multiplying with \(45^n\) makes the product end with the digit 0.

Set 430/4/1

12. Divisibility Check

2 Marks

Check whether \[ 15^n\times2^n \] where \(n\) is a natural number, ends with the digit zero.

Set 430/5/1

13. Prime Factorisation

2 Marks

Using prime factorisation, find the HCF of 180, 140 and 210.

Set 430/6/1

IV

Short Answer Type Questions

14. Irrationality Proof

3 Marks

Prove that \(\sqrt2\) is an irrational number.

OR

Prove that \(\sqrt3\) is an irrational number.

OR

Prove that \(\sqrt5\) is an irrational number.

Sets 430/1/1, 430/2/1, 430/3/1

15. Figure Related – Factor Tree

3 Marks

The factor tree of a number \(x\) is shown in the question paper.
Find the values of \(x,\;y,\;a\) and \(b\).
Hence, write the product of the prime factors of the number \(x\) so obtained.

Set 430/1/1

16. Fundamental Theorem of Arithmetic

3 Marks

State the Fundamental Theorem of Arithmetic.
Use it to find the LCM of 36 and 54.

Set 430/2/1

17. Composite Number Check

3 Marks

Find which among the following numbers \(a,\;b\) and \(c\) is/are composite numbers:

\[ a=7\times11\times13+13 \] \[ b=6\times5\times4+4 \] \[ c=7\times13+6 \]

Set 430/3/1

18. Smallest Divisible Number

3 Marks

Find the smallest number which, when increased by 20, is exactly divisible by 72, 90 and 150.

Set 430/5/2

19. Measuring Rods

3 Marks

Three measuring rods are of lengths 120 cm, 100 cm and 150 cm.
Find the least length of a fence that can be measured an exact number of times, using any of the rods.
How many times will each rod be used to measure the length of the fence?

Set 430/6/1

REAL NUMBERS

Chapter 1 • Basic Mathematics • 2025 PYQs